Dissipative Generation of Spin Squeezing with Multilevel Atoms: Beyond the Bad Cavity Limit

We study a protocol for the dissipative generation of spin squeezing with multilevel atoms in an optical cavity via photon-mediated interactions. We demonstrate that for a broad regime of cavity parameters that go beyond the bad cavity limit, it is possible to generate robust spin squeezing in the full multilevel system and transfer it to the ground state manifold by turning off the drive with no additional pulses. We show that there is minimal deterioration of the squeezing due to the system's strong symmetry, a particular type of symmetry that can emerge in open quantum systems. We verify these capabilities via dissipative phase space methods which use Monte Carlo sampling and stochastic noise to properly capture the build up of quantum correlations in dissipative systems. These methods allow us to explore the parameter regimes where analytical approaches break down under conditions that are relevant to current cavity-QED experiments and are useful for the next generation of atomic clocks.